Images acquired by an imaging system are not perfect. One common type of distortion is shading error (systematic intensity variation across acquired images), which is mainly caused by:    i. Photo-electronic conversion (sensors) inhomogenity;    ii. Illumination inhomogenity;    iii. Optical inhomogenity;    iv. Electronic inhomogenity.
If not corrected, shading error will distort quantification performed from image intensities. It will also distort the visual appearance of the acquired images.
In the prior art, shading errors and shading correction coefficients of imaging systems are calculated from the image of a calibration standard (or reference) with uniform reflectance, transmittance, or emission across the entire field of view. Typically, the materials in a calibration standard are the same or have the same characteristics of the materials to be imaged. For example, to correct a fluorescent imaging system, a slide covered by a thin layer of fluorescent material with uniform thickness and density may be used as a calibration standard. The fluorescent substance in the calibration slide should have the same excitation/emission properties as those to be imaged and/or analyzed subsequently.
Typically, an imaging system is used to image and/or to analyze objects in a continuous fashion. That is, every point in the field of view is of interest. In this case, pixel-by-pixel shading corrections should be performed. FIG. 1 illustrates the mapping between object plane and image sensor plane. The rectangular area 200 in the object plane of an imaging system 250 is projected to the image sensor 202 of the imaging system 250. The area 200 is also referred to as the field of view of the system 250. In the example shown in FIG. 1, the image sensor 202 consists of 48 sensor elements 206 (or pixels) in an 8×6 array, denoted as pxl(x, y), x=1, . . . , 8 and y=1, . . . , 6. Correspondingly, there are 8×6 divisions 205, denoted as Rp(x, y), x=1, . . . , 8 and y=1, . . . , 6, in the field of view 200 in the object plane. The region Rp(x, y) is projected and focused to sensor element pxl(x, y) through the lens 201 of the imaging system 250. In this example, there will be 8×6 shading correction coefficients, each of which is an integrated (average) coefficient of the corresponding region.
If the image sensor of an imaging system has NP by MP elements, and ICP(x, y) is the image pixel intensity of pixel pxl(x, y) of an uniform calibration standard, the pixel-by-pixel shading error SEP(x, y) is defined in EQ. 1. Typically, shading errors are normalized so that the average of all errors equals to one.                                                                         SE                P                            ⁡                              (                                  x                  ,                  y                                )                                      =                                                            IC                  P                                ⁡                                  (                                      x                    ,                    y                                    )                                                                              1                                                            N                      P                                        ×                                          M                      P                                                                      ⁢                                                      ∑                                          i                      =                      1                                                              N                      P                                                        ⁢                                                            ∑                                              j                        =                        1                                                                    M                        P                                                              ⁢                                                                  IC                        P                                            ⁡                                              (                                                  i                          ,                          j                                                )                                                                                                                          ⁢                                          ⁢                                                    For                ⁢                                                                  ⁢                x                            =              1                        ,            …            ⁢                                                  ,                                                            N                  P                                ⁢                                                                  ⁢                and                ⁢                                                                  ⁢                y                            =              1                        ,            …            ⁢                                                  ,                          M              P                                      ⁢                                                      EQ        .                                  ⁢        1            
Typically, an imaging system is a linear device. That is, the raw pixel gray levels of the imaging system have a linear relationship with the input signal. For example, when imaging an illuminant object, the input signal is the self-emitted light intensity from the object. In the linear case, image intensity ICP(x, y) is the raw pixel gray-level of the pixel pxl(x, y) minus a stored background value of the corresponding pixel. In the non-linear case, the non-linear relationship first is modeled, and then this non-linear model is used in calculating the imaging intensity ICP(x, y) from the raw gray-levels so that the image intensity ICP(x, y) is linear with the input signal. If the image intensity is not linear to the input signal or the background values are not properly removed, the calculated shading error will not be accurate. In such a system, a skilled person will be capable of adjusting the image intensity values so that they have a linear relationship with the input signal.
FIG. 2 generalizes the pixel-by-pixel shading error concept. Instead of establishing shading correction coefficients of a region corresponding to each single pixel for the entire field of view, shading correction coefficients can be established for regions corresponding to more than one pixel. The pixel-by-pixel example of FIG. 1 is a special case of this generalized model.
In the example shown in FIG. 2, there are 3×2 regions 215, denoted as R(x, y): x=1, 2, 3 and y=1, 2. Each of these regions projects to its corresponding pixel set 216, denoted as P(x, y): x=1, 2, 3 and y=1, 2, in the imaging sensor 202. Each pixel set in this example contains four pixels. For example, P(1, 1), corresponding to R(1, 1), consists of: {pxl(1, 1), pxl(2, 1), pxl(1, 2), pxl(2, 2)}. The generalized shading correction has 3×2 coefficients, and each coefficient corresponds to an area of 4 pixels. Of course, the regions do not have to be in a regular 2-dimensional matrix form. They can be in any form. For the convenience of discussion, the regular 2-dimensional matrix form is used in this document. The calibration standard may be 1-dimensional or 3-dimensional or may have a non-rectangular 2-dimensional form.
Given a calibration standard with uniform regions R(x, y): x=1, . . . , N and y=1, . . . , M, the shading error SE(x, y) for the region R(x, y) (or pixel set P(x, y)) is defined in EQ. 2.                                           SE            ⁡                          (                              x                ,                y                            )                                =                                    IC              ⁡                              (                                  x                  ,                  y                                )                                                                    1                                  N                  ×                  M                                            ⁢                                                ∑                                      i                    =                    1                                    N                                ⁢                                                      ∑                                          j                      =                      1                                        M                                    ⁢                                      IC                    ⁡                                          (                                              i                        ,                        j                                            )                                                                                                          ⁢                                  ⁢                                            For              ⁢                                                          ⁢              x                        =            1                    ,          …          ⁢                                          ,                                    N              ⁢                                                          ⁢              and              ⁢                                                          ⁢              y                        =            1                    ,          …          ⁢                                          ,          M                                    EQ        .                                  ⁢        2            where: IC(x, y) is the image intensity of region R(x, y). It is calculated by averaging the image pixel intensities of all the pixels corresponding to region R(x, y).
Once the shading error is defined, the shading correction coefficient SC(x, y) for the region R(x, y) is defined in EQ. 3.                                           SC            ⁡                          (                              x                ,                y                            )                                =                      1                          SE              ⁡                              (                                  x                  ,                  y                                )                                                    ⁢                                  ⁢                                            For              ⁢                                                          ⁢              x                        =            1                    ,          …          ⁢                                          ,                                    N              ⁢                                                          ⁢              and              ⁢                                                          ⁢              y                        =            1                    ,          …          ⁢                                          ,          M                                    EQ        .                                  ⁢        3            
The shading correction coefficients SC(x, y) will then be stored and applied to subsequent images to compensate for the system's shading error.
The key requirement of the prior art is uniformity. However, uniform calibration standards are often difficult or impossible to prepare, and, therefore, the resulting estimated shading correction coefficients are often inaccurate.
Another method for calculating the shading correction coefficients is using calibration standards with non-uniform but known local reflectance, transmittance or emission values. Again, it is often very difficult to accurately determine the required reflectance, transmittance or emission values at all the required locations, especially in microscopic imaging applications.